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be an acute-angled scalene triangle with circumcircle 
and circumcenter 
. Suppose 
. Let 
be the orthocenter and 
be the incenter of triangle . Let 
be the midpoint of the arc 
of the circumcircle of triangle 
, containing .
be a point on the arc 
of not containing 
, such that 
. Let 
be the circumcenter of triangle 
. Prove that the lines 
and 
meet on .
be a positive integer. In an 
table, an upright path is a sequence of adjacent cells starting from the southwest corner to the northeast corner such that the next cell is either on the top or on the right of the previous cell. Find the smallest number of grids one needs to color in an table such that there exists only one possible upright path not containing any colored cells.
. Let 
and 
be the feet of the altitudes from 
to and from 
to 
respectively.
and 
as the reflections of across lines and , respectively. Let be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with . is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
) and you also need to be quite good at algebra. You should start with basic euclidean though and perhaps learn inversion first before trying bashing, you will build the fundamentals you need that way.
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\text{Given that } 0 < \theta < 90^\circ,\ \text{solve the equation: } \sin(\theta - 60^\circ)\sin\theta + \sin(54^\circ - \theta)\sin 54^\circ = 0
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\text{What is the value of } \theta\ (\text{in degrees})\ \text{that satisfies the equation?}
\]](http://latex.artofproblemsolving.com/0/0/f/00fbcd5839fedced0e5636c1def5360d65fc0444.png)
